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innovest(x,pdq:vector(p,d,q) [,maxlag:m] [,degree:degree]\
  [,polish:T,cycles:nc] [,checkroots:F] [,silent:T] [,arsign:Arsign]\
  [,masign:Masign]), integers p>=0, d >= 0, q >= 0, maxlag >= p+q,
  degree, nc >= 1, Arsign and Masign = +1 or -1.
Value is structure(phi:phihat, theta:thetahat, xtxinv, nobs:n, npar,
  rss:residSS, neg2logL:-2*log(likelihood), aicc:AICC)

Keywords: arima models, time domain, preliminary estimation
innovest() is a macro to compute the innovations preliminary estimate of
coefficients for an ARIMA(p,d,q) time series as described on pp 151-153
of Brockwell and Davis.  Keywords 'arsign' and 'masign' allow you to
specify the sign convention to be used in defining parameters.  See

innovest(x, pdq:vector(p,d,q) [,maxlag:M]), where x is a REAL vector
and p, d and q are nonnegative integers return a structure summarizing
the results of the preliminary innovations estimates for an
ARIMA(p,d,q) model fit to x.  M >= p + q is an integer with default
value p + q + max(15, p + q).  See below for the form of the results.

There is currently no provision for estimating seasonal ARIMA models

innovest() creates the following side effect variables
  COEF = vector(phihat,thetahat), the estimated AR and MA coefficients
  ALLRESIDUALS = residuals from fitted model including backcast
  NEG2LOGL = -2*log(likelihood) using the estimates found
  NPAR = p + q + degree + 1 = number of coefficients estimated

There are several optional keyword phrases which affect what innovest()
  degree:d1      A polynomial trend of order d1 is removed (after
                 differencing when d > 0).  Nothing is removed, not
                  even a mean, when d1 < 0.  The default for d1 is
                  -d (mean removed when d = 0, nothing otherwise).
  polish:T       The estimates are adjusted by one or more cylcels
                 of an approximate iteration in the direction of the
                  least squares estimates.
  cycles:nc      nc > 0 an integer, default 1 is the number of
                 "polishing" cycles.
  checkroots:F   suppresses checking for stationarity and invertability
  silent:T       suppress warning messages
  arsign:Arsign  +1 or -1; alters definition of AR paramaters; see below
  masign:Masign  +1 or -1; alters definition of MA paramaters; see below

innovest() returns as value
  structure(phi:phihat, theta:thetahat, nobs:n, xtxinv:xtxinv,
    npar:p+q+d1+1, rss:residSS, neg2logL:-2*log(likelhihood),aicc:aicc)

phihat and thetahat are the estimated AR and MA coefficients (NULL when
p or q is 0).

xtxinv is NULL without polish:T or when p = q = 0, Otherwise xtxinv is
an analogue of the regression solve(X' %*% X) derived from the final
polishing step.  Its diagonal elements can be used to compute
approximate standard errors of the coefficients.

residSS = sum of squares of all residuals, including those backcast.

The likelihood is the normal likelihood computed using backcasting and
includes a factor of (2*PI)^(-n/2) and aicc is a modification of
Akaike's information criteria (AIC).

Note that innovest() does not return a mean or other estimates of
detrending parameters.

By default, estimated AR and MA coefficients are checked to see if they
define stationary (causal) and invertable operators, respectively.  If
they do not, a warning message printed and any "polishing" cycles (see
below) are suppressed and components rss, neg2logL and aicc of the
result are set to MISSING.

See topic 'MASIGN' for information on how Arsign (default -1 or the
value of ARSIGN if it exists) and Masign (default -1 or the value of
MASIGN if it exists) modify the definitions of the AR and MA parameters.

To use the convention used by Brockwell and Davis before you start the
analysis, you should use
  Cmd> MASIGN <- 1; ARSIGN <- -1

The convention used by Box and Jenkins (the default) corresponds to
Arsign = -1, Masign = -1.

See also arima(), hannriss(), innovations().

Gary Oehlert 2003-01-15